K-Theory and Structural Properties of $C^*$-Algebras Associated with Relative Generalized Boolean Dynamical Systems

We present an explicit formula for the $K$-theory of the $C^*$-algebra associated with a relative generalized Boolean dynamical system $(\CB, \CL, \theta, \CI_\af; \CJ)$. In particular, we find concrete generators for the $K_1$-group of $C^*(\CB, \CL, \theta, \CI_\af; \CJ)$. We also prove that every gauge-invariant ideal of $C^*(\CB, \CL, \theta, \CI_\af; \CJ)$ is Morita equivalent to a $C^*$-algebra of a relative generalized Boolean dynamical system. As a structural application, we show that if the underlying Boolean dynamical system $(\CB, \CL, \theta)$ satisfies Condition (K), then the associated $C^*$-algebra is $K_0$-liftable. Furthermore, we deduce that if $C^*(\CB, \CL, \theta, \CI_\af; \CJ)$ is separable and purely infinite, then it has real rank zero.